Nonlinear Viscoelasticity and Shear Localization at Complex Fluid

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Nonlinear Viscoelasticity and Shear Localization at Complex Fluid Interfaces Philipp Erni* and Alan Parker Firmenich SA, Materials Science Department, Corporate Research Division, Meyrin 2, Genève, Switzerland ABSTRACT: Foams and emulsions are often exposed to strong external fields, resulting in large interface deformations far beyond the linear viscoelastic regime. Here, we investigate the nonlinear and transient interfacial rheology of adsorption layers in large-amplitude oscillatory shear flow. As a prototypical material forming soft-solid-type interfacial adsorption layers, we use Acacia gum (i.e., gum arabic), a protein/polysaccharide hybrid. We quantify its nonlinear flow properties at the oil/water interface using a biconical disk interfacial rheometer and analyze the nonlinear stress response under forced strain oscillations. From the resulting Lissajous curves, we access quantitative measures recently introduced for nonlinear viscoelasticity, including the intracycle moduli for both the maximum and zero strains and the degree of plastic energy dissipation upon interfacial yielding. We demonstrate using in situ flow visualization that the onset of nonlinear viscoelasticity coincides with shear localization at the interface. Finally, we address the nonperiodic character of this flow transition using an experimental procedure based on opposing stress pulses, allowing us to extract additional interfacial properties such as the critical interfacial stress upon yielding and the permanent deformation.

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comprehensive summary of the development of this field is provided in a review article by Hyun et al.29 High-sensitivity Fourier transform (FT) rheology has dominated the field since the late 1990s following the adaptation of signal analysis techniques from NMR spectroscopy to rheometry by Wilhelm and co-workers.30,31 The current focus is on new quantitative measures to characterize nonlinear properties of soft materials efficiently and interpret them within a physically sound framework.32−36 Cho et al.37 suggested a geometrical approach to waveform analysis based on the decomposition of the nonlinear stress response into elastic and viscous parts. This method was further refined and integrated into a robust mathematical framework using Chebyshev polynominals by Ewoldt et al.34,36 Their concept provides new quantitative measures for nonlinear viscoelasticity that can be directly linked to the material’s constitutive equations and allows a straightforward physical interpretation. Typical procedures for measuring the nonlinear viscoelasticity discard intracycle startup data and use only those stress values considered to be in a steady oscillatory state. Unfortunately, however, plasticity and irreversible flow transitions might be most evident in these startup points. In the worst case for sensitive soft materials, plastic yielding or irreversible fracture might occur even before the oscillatory data can be analyzed using typical large-amplitude oscillatory shear (LAOS) frameworks. Therefore, in the second part of this

INTRODUCTION The non-Newtonian character of interfacial adsorption layers formed by proteins and other biopolymers,1−4 nano- or microparticles,5−7 and amphiphilic (block) copolymers8 or polymer brushes9 has been known for a long time;10−14 however, precise experimental control of shear stresses at liquid interfaces has only become possible more recently.5,15−18 Recent progress in interfacial rheology now allows the classification of these materials by their rheological characteristics in a quantitative and detailed manner similar to that of bulk soft materials. Most applications where interfacial rheology is relevant involve large interfacial deformations and strong bulk flows.19−22 Interfacial yielding is important for flows and stability criteria of emulsions and foams because flow processes such as liquid drainage from foam lamellae or the shear-induced deformation of emulsion droplets can be strongly influenced by the degree of mobility at the interface.21−23 However, shearinduced flow transitions in adsorbed or spread interfacial layers have received much less attention than yielding phenomena in bulk gels or concentrated suspensions. Various soft-solid interfacial layers described in the literature share some mesostructural characteristics with 3D jammed systems but are otherwise quite diverse in composition. Examples include globular proteins (β-lactoglobulin,13 lysozyme,22 bovine serum albumin, and ovalbumin), ultrathin nanocrystalline films,6 spread layers of silica particles,24 and condensed-phase Langmuir layers.25−27 Nonlinear bulk rheology has been studied in steady, transient, and oscillatory flows of polymers for decades;28 a © 2012 American Chemical Society

Received: March 9, 2012 Revised: April 23, 2012 Published: May 7, 2012 7757

dx.doi.org/10.1021/la301023k | Langmuir 2012, 28, 7757−7767

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transitions using a nonoscillatory transient experiment with a measuring protocol based on opposing stress pulses to obtain both linear and nonlinear parameters in one experiment, including critical stresses and permanent deformations.

article, we will also quantify linear-to-nonlinear transitions in transient experiments, which are better suited to nonperiodic singular events such as fracture, yielding, and irreversible flow transitions. Stress pulse (“creep”) experiments with increasing stress levels have been used previously to measure yielding transitions both in the bulk38 and at interfaces. Jaishankar et al.39 have described inertioelastic oscillations observed in a stress-controlled surface rheometer with a ring geometry. In this case, the relative effects of surface elasticity and instrumental inertia40 were in a range in which the resulting oscillations of the deformation were sufficiently strong to be analyzed using appropriate rheological models as described previously for 3D weak gels by Baravian and co-workers.41−43 Free damped oscillations following a step deformation of the interface have already been described by Tschoegl10 in the 1960s and implemented experimentally by Krägel et al.11,44 Their method uses the frequency and the decay of the deformation amplitude of a freely oscillating disk suspended from a torsion wire to calculate the interfacial shear viscosities and elasticities. Between recent advances in interfacial rheology and nonlinear bulk rheology, there is a knowledge gap in nonlinear interfacial rheology, and the field has not yet caught up with recent advances in the understanding of bulk soft materials under large deformations.29,34,37,45,46 Examples of 2D rheology data obtained at large deformations include a study describing strain rate superpositions (SRS) for sorbitan esters26 and a very recent study on hydrophobins47 wherein stress−strain curves in the large deformation regime were described by fitting a mixed Maxwell/Herschel-Bulkey rheological model to the data. Otherwise, dynamic interfacial shear rheology is mostly still limited to traditional deformation or stress amplitude sweep experiments. In this article, we apply the concepts of nonlinear viscoelasticity in large-amplitude oscillatory shear deformation to interfacial adsorption layers, and we demonstrate that interfacial rheology greatly benefits from an improved understanding of these properties. We focus on interfacial shear deformations and specifically investigate the transient and nonlinear interfacial shear rheology of soft-solid-type adsorption layers. As a prototypical interfacial material of this type, we use Acacia gum,48−50 a protein/polysaccharide hybrid known to form dense adsorption layers at oil/water interfaces. It is widely used49,51−53 in applications such as beverage and flavor oil emulsions, pharmaceutical emulsions, paper, textiles, and carbon black ink and to stabilize carbon nanotubes. Three main fractions have been identified:48,52−54 (i) an arabinogalactan fraction (Mw ≈ 2.79 × 105 g·mol−1); (ii) a highmolecular-weight (M w ≈ 1.45 × 10 6 g·mol −1 ), highly amphiphilic arabinogalactan−protein complex (AGP); and (iii) a minor glycoprotein fraction. Previous studies on the interfacial properties of Acacia gum have focused on the interfacial tension, the surface pressure/area isotherms, the quantification of interfacial shear and dilatational rheological properties,49,52,53,55,56 the interfacial activity of its separate fractions, 57 and the interfacial properties of complex coacervates formed with proteins.58 We analyze the nonlinear stress and deformation data to access various measures of nonlinear viscoelasticity, including the intracycle moduli at maximum and zero deformations and the plastic energy dissipation upon interfacial yielding. To assess the interfacial strain field, we perform in situ flow visualization experiments. Finally, we quantify yielding

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RESULTS AND DISCUSSION Nonlinear Viscoelasticity: Large-Amplitude Oscillatory Shear Flow at Liquid Interfaces. We start from a traditional deformation amplitude sweep experiment for an Acacia gum layer adsorbed at the oil/water interface, as shown in Figure 1. A sinusoidal shear deformation is imposed on the

Figure 1. Typical amplitude sweep experiment for Acacia gum adsorption layers in interfacial shear deformation, measured at the oil/ water interface. Frequency ω = 1 rad·s−1. The interfacial storage modulus G′ and the interfacial loss modulus G″ are equivalent to the first-harmonic Fourier moduli G′1 and G″1. The shaded area emphasizes the region where nonlinearities in the stress−deformation waveforms become relevant, indicating the limit of the linear viscoelastic regime and the region of interest for large-amplitude oscillatory shear flow (LAOS).

sample, and its amplitude γ0 is ramped up logarithmically at a constant frequency ω. For structured materials, the elastic shear modulus G′ and the viscous shear modulus G″ typically remain constant up to a limiting deformation or stress; the range of deformations and stresses below this limit is called the linear viscoelastic regime (LVE), and its upper limit defines the onset of nonlinear viscoelasticity as γ0 is increased further. If a deformation function of the form γ(t) = γ0 sin(ωt) is imposed on a linear viscoelastic material, then the resulting stress response is a also harmonic function τ(t) = |G*(ω)|γ0 sin(ωt + δ) with the same frequency; |G*| is the magnitude of the complex shear modulus, and δ is the phase lag between the stress and the deformation signals. For nonlinear experiments, τ(t) can be described by a Fourier series29,31 (written here using the modulus notation) with only odd harmonics: τ(γ0 , ω , t ) = γ0

∑

[Gn′(γ0 , ω) sin(nωt ) + Gn′′(γ0 , ω)

n = 1,3,..

cos(nωt )]

(1)

Here, the nonlinear contributions are captured in the higherorder harmonics n = 3,5,.... To access the rheological information not contained in the first-harmonic Fourier moduli, we perform cycle-by-cycle experiments, testing each deformation amplitude γ0 at different frequencies ω. In Figure 2, we show the measured interfacial stress waveforms along with the imposed deformation function, represented in matrix form for each (γ0, ω) pair. Figure 3 compares the measured interfacial shear stress τexp(t) with the 7758


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Figure 2. Raw waveforms of the normalized imposed deformation function γ(t)/γ0 and the measured interfacial shear stress τ(t) positioned in a matrix defined by the imposed deformation amplitude γ0 and frequency ω. Normalized deformations are scaled identically in all of the graphs; the measured interfacial stresses are plotted on identical scales within each row, except for the last graph where a different scale was chosen for the stress axis. Inertia-dominated curves for the highest γ0 and ω (notice the phase jump for the waveform plotted in gray) are not used for evaluation.

single-harmonic response τ1(t) as calculated from G′1 and G″1. Also shown are the residuals between the linear assumption and the experimental value over an oscillation cycle. They are

highest at the peak values max(τexp), where the single-harmonic underpredicts the experimental values by almost 40%. The corresponding Lissajous curves are shown in Figure 4 in the elastic representation τ(γ) and the viscous representation τ(γ̇). In the first row, the measured interfacial shear stresses are nearly perfect sinusoidals, as expected for the linear viscoelastic regime. The small phase lag between the deformation and stress signals is mirrored by the predominantly elastic response with G′ > G″ found using traditional linear rheology. Increasing the amplitude causes the waveforms to become more complex in shape, and higher-harmonic Fourier series are needed to approximate the data. (This is the approach traditionally used in FT rheology,29,31 which we do not follow in further detail here.) The graphical representation in the form of Lissajous curves with closed stress−deformation loops (Figure 4) gives a global summary of the nonlinear response. In the linear viscoelastic limit, the Lissajous curves are ellipses, with the slope of their longer axes in the (τ, γ) plane providing the magnitude of the complex interfacial shear modulus |G*|; their width is related to the phase angle between the deformation and stress signals. With increasing deformation amplitude, the

Figure 3. Comparison of the experimental interfacial shear stress τ(t) () and the corresponding waveform obtained from the firstharmonic Fourier component (---) for γ0 = 300% and ω = 6.28 rad·s−1. The difference between the experimental values and the fitted curve along one oscillation cycle is shown as shaded bars. Its magnitude is in the range between |τi,1 − τi,exp| ≈ 10−4 and 2 × 10−3 N·m−1. 7759


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Figure 4. Large-amplitude oscillatory shear loops of an oil/water interface plotted in a matrix of deformation amplitudes γ0 and frequencies ω. (a) Elastic Lissajous curves: the interfacial shear stress τ(t) is plotted as a function of the normalized deformation γ(t)/γ0. (b) Viscous Lissajous curves: τ(t) is plotted as a function of the normalized shear rate γ̇(t)/γ̇0. The inertia-dominated curves for the highest γ0 and ω (bottom right corners of the subpanels; curves plotted in gray) are not used for evaluation but are included only for reference.

G′1 (i.e., the nonlinear measures normalized by the linear firstharmonic modulus). The further these ratios are from 1, the stronger the nonlinearity. Figure 6b shows the map of the zero-strain modulus G′M = dτ/dγ|γ→0. Up to deformation amplitudes of γ0 ≈ 30%, this measure is roughly equivalent to G′1 and does not reveal much additional information. In particular, G′M changes with the frequency in parallel to G′1. The underlying τ(γ) curves remain close to an elliptical shape (Figure 4), and the first-harmonic approximation describes the viscoelasticity reasonably well in this regime. The zero-strain modulus remains similar to G′1 even in an intermediate regime where the traditional amplitude sweep test (Figure 1) has already surpassed the LVE regime. Seen differently and taking into account the definition of G′M, traditional strain amplitude sweeps of the adsorption layers studied here provide a relatively accurate picture of the local interfacial shear response near the zero-strain (or maximum strain rate) portion of each oscillation cycle as long as the deformation amplitude is only moderately high. However, as the deformation amplitude is raised further, G′M increasingly deviates from G′1. Darker colors in the map for G′M indicate that the individual Lissajous curves become flatter in the center near γ(t)/γ0 → 0. The actual intracycle, small-strain elasticity is therefore much smaller than would be predicted from G′1.

loops become more complex and nonellipsoidal: simple linear approximations reflected in the traditional G′ and G″ either overpredict or underpredict the real shape and obscure much of the physical information evident in the original waveforms. In Figure 6, we summarize the nonlinear interfacial shear rheology of Acacia gum adsorbed at the oil/water interface in contour maps according to their (γ0, ω) coordinates. (This representation is also called a Pipkin space;34,60 here, we use ω−1 instead of the frequency because for our data we found this scaling to show most clearly how the nonlinear measures change across the parameter space.) We focus on the elastic measures obtained in interfacial shear flow. The map of G′1, the first Fourier coefficient of the elastic interfacial shear modulus, provides a global picture and generalizes simple amplitude sweep graphs to a representation that includes both frequency and deformation. This first map can be obtained without any further data analysis and without a knowledge of the raw waveforms because G′ is equivalent to G′1 (as obtained from standard rheometry software) and it provides a rough picture of the overall changes in the level of the interfacial shear modulus with frequency and deformation. However, it does not contain any information about the important intracycle nonlinearities. To access these properties, we plot two different nonlinear measures,34,36 G′M and G′L (Figure 5) in the same (γ0, ω−1) space. To emphasize the additional information gained with respect to linear methods, we plot the ratios G′M/G′1 and G′L/ 7760


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Figure 5. Schematic of the measures used here to quantify intracycle nonlinear viscoelasticity following Ewoldt et al.34,36,59 G′M: zero-strain modulus obtained by the differentiation of the τ(γ) loop at γ → 0. G′L: large deformation modulus defined from the secant value as γ → γ0. The scheme also illustrates the definition of the plastic dissipation ratio ϕ = Ed/Ep = ∫ τ dγ/4γ0τmax. Ed is the energy dissipated within a single cycle and corresponds to the area inside the Lissajous curve28 (in white); Ep is the dissipated energy expected for a perfectly plastic material, corresponding to the area of the shaded rectangle demarcated by the points (−τmax, −γ0) and (+τmax, +γ0), where τmax is the maximum stress and γ0 is the deformation amplitude.

The large-strain modulus G′L = τ/γ|γ=±γ0 is shown in Figure 6c. In this case, the ratio G′L/G′1 strongly depends on both γ0 and ω. Along with the values of G′M, this measure indicates that the Lissajous curves become increasingly box-shaped, gradually approaching the ideal plastic response corresponding to a rectangular loop. The integration of the interfacial stress over the closed path of the Lissajous curves yields a quantity with dimensions of an energy per unit area, Ed = ∫ τ dγ, associated with the energy dissipated along an oscillation cycle.28 A very useful concept for assessing plastic dissipation in soft bulk materials was recently introduced by Ewoldt:59 Ed is normalized by the dissipated energy expected theoretically for a perfectly plastic material Ep = 4γ0τmax (which is the area of a Lissajous “box” associated with a perfectly plastic oscillatory response), resulting in a dissipation ratio ϕ = Ed/Ep = ∫ τ dγ/4γ0τmax. ϕ → 1 means that the material is perfectly plastic; ϕ → 0 is an entirely elastic response (in which case the Lissajous figure approaches a line with slope G). For all experiments shown here, ϕ is below 1 and is smaller than the Newtonian reference case59 (ϕ ≈ π/4). Within a classification scheme for structured fluids, this type of behavior therefore combines plastic, elastic, and viscous features. When we compare the map for ϕ to those for G′1, G′M, and G′L, the plastic dissipation ratio is the most sensitive parameter for shear-induced changes in the adsorption layers studied here. To obtain the nonlinear measures, we do not make use of higher-order harmonics of a Fourier transform,31 Chebyshev polynomials,34 or the a priori assumption of constitutive models.35,47 Instead, we numerically extract these parameters by differentiation of the τ(γ) data at γ → 0 for G′M and by calculating the secant value on τ(γ) as γ → γ0 for G′L. This simplified approach relies only on parameters determined directly from the τ(γ) curves. Figure 6a−c also emphasizes the difference between the geometric or polynomial approaches to nonlinear viscoelasticity

Figure 6. Contour plots of the nonlinear measures for the interfacial shear modulus of Acacia gum adsorbed at the oil/water interface. (a) (γ0, ω−1) space of the first-harmonic Fourier coefficient of the interfacial shear elastic modulus G′1 (equivalent to G′); this first map can be constructed from simple deformation amplitude sweep experiments at different frequencies without using nonlinear measures. (b) Zero-strain modulus G′M, defined by the slope of the τ(γ) Lissajous curve for γ → 0 (Figure 5). (c) Large strain modulus G′L for γ → γ 0.

and the strain rate superposition61 (SRS) method: using the latter, no higher harmonics are accounted for (meaning that we would obtain G′M = G′L = G′1). Subpanel a therefore contains information similar to that obtained in SRS experiments, whereas subpanels b and c quantitatively summarize the information gained by accounting for intracycle nonlinearities. These plots therefore also reveal the regions where results from G′1-based SRS need to be interpreted with care. In the experiments performed here, we set a custom-defined, interval-wise deformation using the standard closed-loop control system62 in an instrument that is essentially torque7761


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controlled on the hardware level. For the calculations of the nonlinear elasticity measures provided below, this does not introduce a significant error and the method is robust, as indicated by the quality of the imposed deformation sine functions. The advantage of this method is that no specific equipment, software other than that used for standard data processing, or additional data acquisition hardware is needed. A more difficult problem is the influence of inertia upon increasing γ̇. An example is the experiment in Figure 2 for the largest deformation amplitude; we do not use this type of Lissajous figure for analysis. Nevertheless, we include the graph here as a reference to a “pathological” case to indicate potential experimental pitfalls, in this case due to the interface having yielded, with the consequence that subphase viscosity and inertia dominate the measured response. Inertia is, however, not always an undesired effect in interfacial rheology. Indeed, when properly analyzed, the coupling between inertia and the elasticity of the material can be used to extract useful information and is an alternative way of quantifying viscoelastic properties. Inertioelastic coupling has been known for a long time but is not frequently used to analyze material properties. Starting with the work of Baravian and co-workers41−43,63 on the rheology of weak gel systems, these effects have received renewed attention lately.39,64 Here, we will use this approach to complement the large oscillatory deformation experiments with transient data more suitable for quantifying nonperiodic flow transitions. Yielding Transitions of Interfacial Adsorption Layers. Densely packed adsorption layers can resist lateral stresses and remain stagnant even if the subphase liquid flows. However, they exhibit strongly nonlinear mechanical behavior as the stresses and deformations increase. This is analogous to typical yield stress fluids in 3D,65 including gels, pastes, colloidal glasses, and concentrated surfactant solutions. In the classical picture of a Bingham model fluid, these behave as solids at small stresses but flow like liquids above a critical stress. Interestingly, the question of whether the yielding is reversible is often ignored. Different yielding processes may share many rheological features but can have quite different physical origins. For example, concentrated dispersions of microgel particles possess thixotropic restructuring times on the order of milliseconds66 and quickly recover their mechanical properties after being exposed to a large deformation, whereas for many pastes or gels yielding is associated with plastic deformation and permanent microstructural changes.67,68 In the following text, we assess which of the two pictures is more relevant for rigid but soft biopolymer adsorption layers. In Figure 8, we show a visualization of the deformation field of Acacia gum at the oil/water interface for both the linear viscoelastic (deformation amplitude γ0 = 5%) and the nonlinear (γ0 = 300%) regimes. We trace silica particles spread at the interface using video image analysis. In the linear regime, the interfacial deformation looks as expected for simple shear flow, with a linear surface velocity profile from the outer edge (zero velocity on the surface) to the inner edge (maximum surface velocity equivalent to the velocity at the edge of the moving disk). No discontinuities in the deformation field are observed, and there is no edge slip of the surface against the inner and outer edges. This is in stark contrast to the behavior above the yielding transition, where we observe strong shear localization. In the example shown in Figure 8, the surface separates into shear bands: one with a low average velocity near the stationary outer edge of the flow cell and one with a high average velocity

Figure 7. Interfacial plasticity landscape: the plasticity measure ϕ of Acacia gum adsorbed at the oil/water interface is shown as a function of the deformation amplitude γ0 and the frequency ω. The case ϕ → 0 indicates a purely elastic interface; ϕ → 1 is a perfectly plastic 2D solid. The maximum value found here is ϕ = 0.716.

near the moving disk. Therefore, the apparent yielding process of the interfacial layer is not a global “unjamming” event, but it is a flow transition from homogeneous simple shear flow toward strong shear localization and the coexistence of high shear zones at the edges and a low shear zone in the middle of the surface. Oscillatory experiments do not always provide a full picture of yielding, plastification, or fracture phenomena that result in the partial destruction of the sample. In particular, startup effects occurring within the first oscillation cycles are usually discarded. These tests should therefore be complemented with transient experiments performed in the time domain. Moreover, the notion of a yielding transition can be ambiguous and has been used alternatively for systems with very fast restructuring (e.g., concentrated suspensions of microgel particles) and for materials where the yielding process is irreversible and associated with permanent structural changes. The scalar measure ϕ does not directly address those differences. To assess nonperiodic shear yielding events, we use a protocol based on opposing stress pulses (OSP, Figure 9). In these sequential creep experiments, we impose a constant interfacial shear stress and measure the resulting deformation response (creep curve). At the end of each stress pulse, we set the stress to zero and keep tracking the deformation (recovery curve). To minimize the total strain experienced by the sample, each forward step is followed by a backward step with a stress pulse of identical magnitude but opposite sign. Subsequently, we increase the stress level, following the same protocol. The result is an amplitude sweep experiment where the imposed function of the interfacial shear stress is a step function rather than a sinusoidal, with sequential, constant values of the interfacial stress τ(t) = +|τ0| and τ(t) = −|τ0| (each of identical length Δt), where the opposing steps are separated by a recovery interval (τ(t) = 0) and the level of the stress pulses τ0 increases from one opposite pair of steps to the next. The OSP test offers the advantage of keeping the accumulated strain as low as possible as the nonlinear regime is approached (similar to a harmonic amplitude sweep experiment) while at the same time providing access to nonperiodic events on the time scale of a single cycle. In addition, semifree oscillations caused by the interplay of inertia and sample elasticity may be promoted or suppressed at will by 7762


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Creep curves with inertioelastic oscillations (ringing) can also be used to obtain additional viscoelastic parameters either by measuring the natural oscillation frequency and the damping of the amplitude of the free oscillation or by the numerical fitting of an appropriate constitutive equation.39,41,63,69 An example for a nonlinear least-squares fit of a Maxwell−Jeffreys model41 to a creep curve as well as the envelope for the amplitude decay of the free oscillations is shown in Figure 10a. Using Struik’s method40 to approximate the interfacial shear moduli, we obtain a complex modulus of |G*| ≈ 0.011 N·m−1 and a loss tangent of tan(δ) = G″/G′ ≈ 0.22 at a natural frequency of ω0 = 2.82 rad·s−1, which is in reasonable agreement with |G*| = 0.015 N·m−1 and G″/G′ = 0.21 found in forced oscillations under similar conditions. Figure 10b shows the recovery portion of the creep curves (again, only the recovery curves following the forward stress pulses are plotted). Here also, the combination of sample elasticity and inertia result in free, damped oscillations of the deformation. In all cases, the interface does not fully recover within the experimental time window. The permanent deformations γ∞ measured at the end of each recovery step (indicated by the red frame in the figure) can be used as a measure of viscous dissipation if the material is purely viscoelastic. However, as shown above, Acacia gum layers undergo a yielding transition associated with plastic dissipation; therefore, the interpretation of γ∞ as a material property is not straightforward. In the inset of Figure 10b, we plot the permanent deformation as a function of the imposed interfacial shear stress. γ∞ remains small for weak imposed stresses, indicating that elastic recovery dominates. For stresses above τ0 ≈ 4 × 10−3 N·m−1, γ∞ increases more steeply with τ0 than would be expected for purely viscoelastic materials. Interestingly, this value, indicated by the dashed line in the inset, is very close to the maximum oscillatory stress response observed in the Lissajous curves (Figure 4). The critical shear stress at which the interface is no longer able to recover elastically in a creep experiment is therefore very similar to the stress level that the interface self-selects when exposed to a forced sinusoidal strain as described above. At much higher interfacial stresses, the interface has fully yielded, a steady interfacial shear flow develops, and recovery is very small as compared to the accumulated strain (Figure 10d). The interfaces studied here show a global viscoelastoplastic response similar to that of weak bulk gels. However, here elasticity and plasticity are exclusively confined to the interface whereas the subphase liquids are purely viscous. The specific combination of both effects appears to be a rather peculiar phenomenon observed specifically for viscoelastic interfaces on viscous substrates: in three dimensions, similar behavior can be observed in bulk molecular gels with a viscosity-enhanced solvent phase, for example, in gelatin/oligosaccharide mixtures.70 Finally, measuring viscoelastic properties with a stresscontrolled instrument is challenging if the samples are weak gels42 (bulk or interfacial), which in the context discussed here means that they are elastic, but the overall stress levels are relatively low and give rise to strong inertioelastic coupling. Indeed, the spontaneous oscillations observed here can be a significant source of systematic error in frequency sweep experiments; the phase shift and tilted stress waveform evident in Figure 2 (case γ0 = 300%, ω = 12.57 rad·s−1) is an example of this effect.

Figure 8. Visualization of the interfacial strain field in 2D Couette− Searle flow in the linear (LVE) and nonlinear (NLVE) viscoelastic regimes during a startup oscillation cycle (Acacia gum at the oil/water interface; the top edge is the rotating disk). Shown are color-coded overlay images of traced particles. (Top) LVE with frequency ω = 6.28 rad·s−1 and deformation amplitude γ0 = 5%. (Bottom) Interface in the NLVE at large deformations, with ω = 6.28 rad·s−1 and γ0 = 300%, demonstrating strong shear localization at both the inner and outer edges of the geometry. Numbers identify selected points on the surface to guide the eye. γ = 0 shows the interface at zero deformation, γ = γ0 is the image taken as the deformation amplitude value is reached. γ1,2,3 identifies different deformations during the cycle. The scale bar is 1 mm. (Inset) Setup used for flow visualization. (a) CCD camera, (b) prism, (c) fiber light source, and (d) interfacial shear cell and rheometer.

choosing a measuring geometry with either a high or a low moment of inertia (Figure 10a vs c) . Figure 10a shows the creep response under different levels of the applied interfacial shear stress (where each step is followed by a backward step to minimize the total strain in the system, but only the forward steps are plotted here). As the stress is increased, the curves evolve toward behavior dominated more strongly by viscous effects. The magnitude of the creep compliance J(t) = γ(t)/τ0 depends on the imposed stress pulse τ0, indicating that the response is already nonlinear. This behavior can be summarized in more compact form by plotting the peak value Jmax as a function of τ0 (left-hand inset in Figure 10a). The compliance becomes much more sensitive to incremental changes in the interfacial stress as τ0 increases. 7763


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Figure 9. Schematic of the opposing stress pulse (creep) experiments and the resulting deformation response.

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CONCLUSIONS The concepts of nonlinear viscoelasticity in general and largeamplitude oscillatory shear flow in particular are fully applicable and relevant to the characterization, physical interpretation, and analysis of interfacial adsorption layers. We have shown that nonlinear measures previously applied to only the bulk rheology of structured fluids provide a wealth of information on complex oil/water and air/water interfaces under interfacial shear flow. Traditional amplitude sweep and strain rate superposition data are often rather generic for a wide range of materials (bulk or 2D). In contrast, the full nonlinear maps used here are far more sensitive for specific material characterization and can be used for “fingerprinting”59 interfacial adsorption layers, for example, to screen amphiphiles from natural sources or for the formulation of foams and emulsions. We expect our results on the interfacial shear yielding of Acacia gum to be relevant and directly applicable to adsorbed layers of globular proteins,71,72 protein/surfactant mixtures,73 particle layers,7,24 adsorbed hydrophobic bacteria,74 and condensed surfactant Langmuir films,25 where the yielding is caused by fracture events in the layer.75 For example, fracture stresses in globular protein layers are also important during isotropic expansion flows in the overflowing cylinder geometry.76 Shear localization is observed not only in oscillatory shear flow but also in creep tests under high stresses and during the startup of steady shear flow. The geometrical approach based on Lissajous curves is not limited to the specific interfacial measuring geometry used here: stress vs deformation and deformation rate plots at varying frequency and stress or strain amplitude can be generated for other measuring principles in interfacial rheometry, including oscillating needle,15 ring,5 and disk77 devices. In particular, although we perform deformation-controlled experiments, the various frameworks for the analysis of LAOS can also be adapted to experiments where a harmonically oscillating stress is imposed on the interface and the deformation is measured, following the guidelines provided for the bulk case.62 Most current rheometers, including the one used here, are based on a top-driven design combining both the drive motor and the sensing unit. On the hardware level, these instruments are stress (or torque)-controlled. The feasibility of deformation-

controlled experiments done in torque-controlled instruments largely depends on the closed-loop feedback control scheme available. We note that with some top-driven, torque-controlled rheometers (e.g., older instruments with classical drag cup motors) it might be challenging to control the deformation properly. In that case, it is recommended to switch the method to a stress-controlled formulation (i.e., to impose the stress τ(t), measure the deformation γ(t), and use compliances J′ and J″ instead of the moduli62). Finally, large deformation experiments are also important for the related field of interfacial dilatational rheology; although small deformation area oscillations are now routinely used to assess constitutive models for the dilatational modulus,78 largedeformation, nonlinear dilatational experiments can provide further insight into the rheology of interfaces far from equilibrium.79 Moreover, the shear viscoelasticity of liquid interfaces has been shown to become relevant even in pure compressional or dilatational deformations for rigid adsorption layers;80 there also, typical deformations are sufficiently large that nonlinear shear properties are expected to play an important role.

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EXPERIMENTAL SECTION

Materials. Acacia gum was obtained from CNI (Rouen, France) and was hydrated overnight in Millipore water under stirring, centrifuged for 10 min at 1000 rpm, and passed through 5 μm pore size filters; the pH value is 4.5, and no salt is added. The mass concentration in the final solution is 5 ± 0.1% w/w (as quantified by thermogravimetry). For an in-depth compositional characterization, we refer to a recent article by Renard et al.54 Medium-chain caprylic/ capric acid triglyceride oil (Neobee M5) was obtained from Stepan Chemicals (Northfield, IL). The bulk dynamic viscosity of the aqueous biopolymer solution at 25 °C is 5.9 mPa·s; for the medium-chain triglyceride (MCT) oil used as the hydrophobic phase, the bulk viscosity is 25 mPa·s. In the absence of the adsorbed layers, the oil/ water interface exhibits no change in interfacial tension (verified independently using a DSA 10 Mk2 pendant drop tensiometer, Krüss, Germany), and no interfacial rheological response is detectable. The bulk rheological properties of all liquids were measured in a Physica MCR 300 rheometer with concentric cylinder geometries (Anton Paar, Ostfildern, Germany). Interfacial Rheology. We use a Physica MCR 300 rheometer (Anton Paar, Germany) equipped with a biconical disk measuring 7764


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moment of inertia of the disk is I = 2.38 × 10−5 N·m·s2. For the creep and opposing stress pulse experiments, we use a smaller, lighter disk with Ri = 25.00 mm and a moment of inertia of I = 1.1 × 10−5 N·m·s2; in this case, the cup radius is Ro = 30.05 mm. Once the biconical disk is mounted, the motor is precision adjusted to map the actual position of the measuring geometry, minimizing vibrational noise under lowtorque conditions. To position the disk at the fresh air/water interface, the normal force transducer of the instrument is used. The oil phase is added via the upper surface of the disk once a stable air/water/solid contact line has been formed and the compressional stress resulting from the positioning procedure has relaxed. Subsequently, the interface is left to age for 320 min. For large-amplitude oscillatory shear (LAOS) experiments, the shear deformation γ varies sinusoidally with time at an angular frequency of ω: γ(t) = γ0 sin(ωt), where t is time and γ0 is the oscillation amplitude. The torque response M(t) is recorded and converted into an interfacial shear stress waveform τ(t) as described previously.16,81 For each frequency and amplitude, we directly set the deformation by entering the numerical values, interval by interval, into the rheometer software. Startup data points are discarded, and every experiment is preceded by several equilibration loops performed at identical frequencies and deformation amplitudes but with a lower time resolution. The resulting stress data are first reduced and smoothed using spline interpolation, and the nonlinear measures are obtained numerically using polynomials fitted either at the zero or maximum deformations. All data analysis is performed using the ProFit 6.1.15 software (Quansoft). We note that the rheometer used is essentially a stress-controlled instrument used in deformationcontrolled mode; because of the fast feedback loop (direct strain oscillation method62), the imposed γ(t) waveforms are highly accurate sinusoidals. We also note that at higher frequencies and strain amplitudes inertia artifacts occur, as expected with a stress-controlled instrument41,42,63 (Figure 2, bottom right subgraphs plotted in gray; without knowing the waveform or Lissajous data and just using standard, first-harmonic G′ and G″ values, such data might erroneously be interpreted as shear-thickening effects). In creep experiments, a constant stress τ0 is imposed on the sample during a defined time, and the deformation response function γ(t) during this stress pulse (creep curve) is measured. Additionally, this experiment provides information about the recovery behavior of the strain after the removal of the stress. The creep compliance J(t) = γ(t)/τ0 is a material function defined from creep experiments as the ratio of the time-dependent deformation and the imposed stress pulse. In interfacial rheology, the creep compliance of the interface J in units of m N−1 is calculated from the ratio of the measured deformation response γ(t) and the imposed interfacial shear stress τ0 in N m−1. All surfaces in contact with the liquids are cleaned consecutively with Hellmanex III (VWR) detergent dissolved in hot water, followed by purified water, isopropanol, and ethanol and are then dried under nitrogen before and after every measurement. Visualization of the Interfacial Strain Field. To visualize the deformation field during 2D shear flow, we image the interface through a 90° prism from above using a CCD camera equipped with a long-distance macrozoom lens. We spread silica particles (Aerosil 200F, Evonik, Germany) and illuminate the flow cell from the side via a fiber optic white-light source. For the visualization, we use a flow cell with a transparent bottom wall and a slightly wider diameter (Ro = 42.0 mm); images taken on a quiescent, particle-free interface are used as the optical background. The video frames are imported into standard image analysis software (NIH ImageJ, available in the public domain). We highlight and color code a few tracer locations along a selected radius on the annular surface manually; the remaining particles are discarded using threshold filtering. We then superimpose the image background and the color-coded tracers at zero deformation before the creep test (γ = 0) and 15 s after the onset of the imposed stress pulse.

Figure 10. Inertioelastic oscillations in the deformation response of Acacia gum at the oil/water interface at different interfacial shear stresses. Here, inertia is deliberately introduced into the measurement by using an alternative disk geometry with a different moment of inertia, and the deformation response γ(t) to a stress pulse τ0 is measured (semifree oscillations). (a) Interfacial creep compliance J(t) for different imposed stress levels. Each data series corresponds to a level of interfacial shear stress. The insets show the value of the first peak as a function of the applied interfacial shear stress (left) and a typical fit of the Maxwell−Jeffreys constitutive equation and the envelope used to quantify the amplitude decay (right). (b) Recovery curves showing the interfacial shear deformation upon removal of the stress. (Inset) Permanent deformation after creep recovery γ∞ as a function of the imposed interfacial stress level. The dashed line indicates the maximum stress level found in the oscillatory Lissajous curves for comparison. (c) Example of a reference creep experiment performed with the standard biconical disk. Inertioelastic oscillations are minimized and are limited to a few startup data points, plotted in gray (which are routinely discarded in creep experiments). (d) Example of full yielding without significant recovery at higher imposed stresses (torque levels are indicated in μNm). geometry16,81−83 (Figure 1). The disk rotates or oscillates under a controlled torque while measuring the resulting angular deflection. Interfacial shear stresses and the coupling factors accounting for viscous subphase drag are calculated using the analysis described in refs 16 and 81 solving the Stokes equations84 in the measuring cell for an annular interfacial layer between two immiscible fluids.85,86 The global interfacial shear deformation is calculated offline from the measured deflection angle along with the appropriate geometry factors. We use two different measuring cells. For most experiments, the radius of the disk is Ri = 34.14 mm, the cup radius is Ro = 40.05 mm, and 7765


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AUTHOR INFORMATION

Corresponding Author

*E-mail: philipp.erni@firmenich.com. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Christophe Baravian (CNRS/Laboratoire d’Energétique et de Mécanique Théorique et Appliquée, Nancy, France) for helpful discussions.

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REFERENCES

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Nonlinear Viscoelasticity and Shear Localization at Complex Fluid

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